Deputy Attorney General Rod Rosenstein should have been more aggressive with his vow to punish people who leak information to the press, even if they are White House staffers or members of Congress, former Education Secretary Bill Bennett said Monday.

"He didn't mention the press," Bennett told Fox News' "Fox and Friends" program. "The First Amendment is not a license to ruin your country, and there are laws that say that the press cannot take the leaks when they harm the country and go publicly."

Bennett said he wishes Rosenstein would have not excluded certain categories with his vow to go after leakers.

"Imagine, that somebody got a hold of a report from Gen. James Mattis directing, you know, where U.S. troops would go after ISIS in Raqqa, should that be reported or not?" Bennett asked. "No, it should not be reported and if somebody does and if we lose lives, that person should be prosecuted. There's no immunity here. Good lawyers defend people. My brother defended Judith Miller who stayed in jail. The law has to be honored."

Also on the program, Bennett said he does believe that Vice President Mike Pence was "truly offended" by reports that he was positioning himself to run for president in 2020.

"I'm pretty close to the vice president and very close to his staff," Bennett said. "He has a wonderful staff. I talked to those people on a semi-regular basis."

Bennett also said Monday he hopes Trump's lawyers are gearing up for a tough fight from special counsel Robert Mueller and his investigators.

"He's hired 10 or 12 of the roughest prosecutors," Bennett said. "The venue is in D.C. He says it's not a fishing expedition. Call it a hunting expedition. These guys are out for bear and it's going to be very rough and I hope the president's lawyers gear up. Trump will be on and who will replace him, part of the narrative because they didn't like the results of the last election."

Deputy Attorney General Rod Rosenstein should have been more aggressive with his vow to punish people who leak information to the press, even if they are White House staffers or members of Congress, former Education Secretary Bill Bennett said Monday.